Estimation and Pricing under Long-Memory

Stochastic Volatility

Alexandra Chronopoulou Frederi G. Viens

Department of Statistics, Purdue University

150 N. University St. West Lafayette

IN 47907-2067, USA.

achronop@stat.purdue.edu viens@purdue.edu (corresponding author)

Abstract

We treat the problem of option pricing under a stochastic volatility model that exhibits

long-range dependence. We model the price process as a Geometric Brownian Motion

with volatility evolving as a fractional Ornstein-Uhlenbeck process. We assume that the

model has long-memory, thus the memory parameter H in the volatility is greater than

0.5. Although the price process evolves in continuous time, the reality is that observations

can only be collected in discrete time. Using historical stock price information we adapt

an interacting particle stochastic filtering algorithm to estimate the stochastic volatility

empirical distribution. In order to deal with the pricing problem we construct a multino-

mial recombining tree using sampled values of the volatility from the stochastic volatility

empirical measure. Moreover, we describe how to estimate the parameters of our model,

including the long-memory parameter of the fractional Brownian motion that drives the

volatility process using an implied method. Finally, we compute option prices on the S&P

500 index and we compare our estimated prices with the market option prices.

Keywords: Option pricing; stochastic volatility; long memory; particle filtering;

multinomial tree.

JEL classification code: C14, C5, C6, G13

1 Introduction

In the celebrated model in option pricing theory, due to Black and Scholes (1973, [3]) and

Merton (1973, [21]), the dynamics of the underlying asset evolve according to a geometric

1

Brownian motion. However, option prices in practice are not conformable with the theoret-

ical assumptions of this model. One of the main points of discussion is the method used

for determining the volatility parameter. The empirical option pricing literature deals with

this problem extensively suggesting several techniques, such as historical or implied volatility.

Black-Scholes implied volatility does not capture the structure of reported option prices and

as a result we observe the smile effect, well documented in the literature, that is the U-shaped

pattern of implied volatility when plotted against different strike prices.

These inconsistencies indicate that the movements of stock prices are not entirely ex-

plained by the basic Black-Scholes (BS) model. One direction to understand them better is to

withdraw the constant volatility assumption, and assume instead that volatility is described

by a stochastic process. Models in this framework are called stochastic volatility models.

Let {St; t ∈ [0, T ]} denote the stock price process with dynamics given by

dSt = µSt + σ(Yt) St dWt, (1)

where σ(·) is a deterministic function and Y is a stochastic process which one can call the

volatility process. Under typical stochastic volatility models, Y is the solution of a stochastic

differential equation (SDE) driven by a noise Z that could be independent of W , or correlated

with it. For example, Y can be described by a

• Log-Normal process: dYt = c1 Yt dt + c2 Yt dZt,

• Mean Reverting Ornstein-Uhlenbeck (OU) process: dYt = α (m − Yt) dt + β dZt,

• Feller or Cox–Ingersoll–Ross (CIR) process: dYt = k (ν − Yt) dt + v√

Yt dZt.

Depending on the volatility process Y and the function σ(·), we have a variety of

models. For instance, the Hull-White model (1987, [18]) has σ(y) =√

y and Log-Normal Y ,

the Scott model (1987, [24]) uses σ(y) = ey and a mean-reverting OU process for Y , the model

suggested by Stein and Stein (1991, [26]) sets σ(y) = |y| and Y again a Mean-Reverting OU

process, or the Heston model (1993, [17]) assumes that σ(y) =√

y and Yt is a CIR process,

but with noise correlated with the Brownian motion that drives the asset prices. More details

about these models can be found in Fouque et al., [14].

In this article, we consider a long-memory stochastic volatility model. That is, we

describe Y by a long-memory process. The idea of long-memory stochastic volatility is not

new in the literature. It has been empirically observed that the autocorrelation function of

the squared high frequency returns is usually characterized by its slow decay towards zero.

This decay is neither exponential, as in short-memory processes, nor implies a unit root, as

2

in integrated processes; see for example Ding et al. (1993, [12]), Dacorogna et al. (1993, [9])

and Lobato and Savin (1998, [19]), among others. Consequently, it has been suggested that

squared returns may be modeled as a long memory process, whose autocorrelations decay at a

hyperbolic rate. In this direction, in discrete time, Breidt et al. (1998, [4]) and Harvey (1998,

[16]) proposed independently the Long-Memory SV (LMSV) model, where the log-volatility is

modeled as an ARFIMA(p; d; q) process.

Comte and Renault (1998, [7]) introduced a continuous-time mean reverting process in

the Hull-White setting. In this way, they were able to explain the persistence of the stochastic

feature of implied volatilities in the Black-Scholes setting, when the time to maturity increases.

Under their fractional stochastic volatility model, the volatility process is described by a frac-

tional Ornstein-Uhlenbeck process; that is the standard Ornstein-Uhlenbeck process where the

Brownian motion is replaced by a fractional Brownian motion.

Following the approach of Comte and Renault, [7], we work in continuous time, and

assume that the dynamics of the volatility are described by

dYt = α Yt dt + β dBHt

where {BHt ; t ≥ 0} is a standard fractional Brownian motion with H > 1/2. Under this model

the volatility exhibits long-memory, meaning intuitively that the volatility today is correlated

to past volatility values with a dependence that decays very slowly. In this way we introduce

long-memory in our model, but not directly in the returns, as was suggested by Cheridito

(2003, [5]), Rogers (1997, [23]) and Sottinen (2001, [25]). On the contrary, we assume that

the returns of the stock price are independent, which is an assumption that is supported by

empirical findings, and also allows us to remain in an arbitrage-free context in continuous time.

The stochastic volatility model (1), with µ being the mean rate of return, is the stock

price under the objective ( market) probability measure, which we denote by P. This is the

model that we use when we deal with historical prices, for statistical inference purposes for

example. However, in order to deal with the pricing problem we are in need of an arbitrage-

free pricing scheme. Thus, we will introduce an equivalent martingale measure denote by Q,

under which the stochastic volatility model is as in (1), but with µ replaced by r, the risk-free

interest rate. The two measures P and Q are equivalent by Girsanov’s theorem. In the sequel

we switch between these two measures appropriately, depending on the context (estimation

or pricing). We will also see that Q is not necessarily unique, i.e. the market is incomplete,

which is consistent with the idea that a model with one risky asset and more than one source

of noise cannot be complete.

3

Our goal in this article is to suggest an option pricing algorithm as well as a parameter

estimation procedure for our long-memory stochastic volatility model. In preparation for the

task of option pricing, we adjust a genetic-type particle filtering algorithm by Del Moral et

al. (2001, [10]) adapted by Florescu and Viens (2008, [13]) in a stochastic volatility setting

with no memory, in order to estimate the empirical distribution of the volatility. In the sequel,

following the approach in [13], we use a multinomial recombining tree algorithm for option

pricing. Using simulated data we show that this algorithm performs well and captures the

underlying memory of the system.

Before applying our method to real market data, we also treat the statistical inference

problem for the suggested model. We observe that the crucial issue is to estimate the long-

memory parameter, H, properly. A parametric approach for estimating H is not an easy task,

since the volatility process is not observed and the long-memory effect can only be seen in

the squared log-returns. We test a variety of non-parametric techniques which turn out to be

quite disappointing. Therefore, we suggest an implied method to compute H by calibrating

our computed option prices for different values of H with the realized option prices.

We apply this procedure using market data from the S&P 500 index IBM stock and we

compare our results with the Binomial model and the BS model using implied volatility. At

the same time, we are able to investigate the existence of long-memory in our data. In general,

it seems that in the S&P 500 there is evidence of long-memory, while there is lack of it for

IBM stock. Moreover, we work with data during the 2008 financial crisis, which started on

Wall Street and propagated to the larger financial system: we study whether market exhibits

long-memory after the successive failiures of Bear Stearns Inc., Lehmann Brothers, and AIG.

The structure of the paper is as follows. In Section 2, we describe the proposed model

and discuss its main properties. Section 3 is devoted to the description of a pricing algorithm for

the proposed model based on a combination of a particle filtering technique with a multinomial

recombining tree algorithm. In Section 4, we treat the estimation problem for the various

parameters of the model, while in the last section we apply our methodology to pricing european

call options written on the S&P 500.

2 Long-Memory Stochastic Volatility Model

We choose to work with the log-returns of the stock, i.e. equivalently, the logarithm Xt = log St

of the price process. Under a given martingale measure, these returns are driven by the

4

following equations{

dXt =(

r − σ2(Yt)2

)

dt + σ(Yt) dWt,

dYt = α Yt dt + β dBHt ,

(2)

where r is the short-term risk-free rate of interest, {Wt; t ≥ 0} is a standard Wiener process

and {BHt ; t ≥ 0} is a standard fractional Brownian motion with H > 1/2, both under the

martingale measure. We assume that W and BH are independent, although this assumption

could be relaxed, in order to account for leverage effects, which will be the topic of future

investigations.

2.1 Fractional Brownian motion

Before discussing the properties of the fractional SDE (2), we discuss some of the properties

of the fractional Brownian motion, which is the driving noise of this process.

Definition 1 A fractional Brownian motion (fBm) with Hurst parameter H ∈ (0, 1] is a

centered Gaussian process {BHt ; t ∈ R+} whose distribution is defined by its covariance

Cov(BHt , BH

s ) =1

2

(

|t|2H + |s|2H − |t − s|2H)

, t, s ∈ R+,

and the fact that its paths are continuous with probability 1.

The covariance of fBm immediately implies that it has H-self-similar increments; for

every c > 0 the processes {BHct ; t ∈ R+} and {cHBH

t ; t ∈ R+} have the same distribution. The

mean square of the increments of fBm computes as

E(

|Bt − Bs|2)

= |t − s|2H , (3)

which directly indicates that the increments are stationary. For H = 12 , the process is standard

Brownian motion (the Wiener process). However, contrary to standard Brownian motion, fBm

is not a semimartingale nor a Markov process when H 6= 12 .

The fBm has one additional very important property for certain values of H: its long-

range dependence (a.k.a. long-memory). Indeed, when H 6= 12 , the increments of fBm over

disjoint intervals are not independent; their correlation function is

ρH(n) =1

2

(

(n + 1)2H + (n − 1)2H − 2n2H)

.

We observe that when H < 12 then ρH(n) < 0 and the increments over disjoint intervals are

negatively correlated. When H > 12 then ρH(n) > 0 and the increments over disjoint intervals

5

are positively correlated. More specifically in the case that H > 12 the stationary sequence Xn

exhibits long-range dependence (or long memory) in the sense that∑∞

n=1 ρH(n) = ∞, which

follows immediately from the asymptotics ρH(n) = H(2H−1)n2H−2+o(

n2H−2)

. When H < 12 ,

then∑∞

n=1 |ρH(n)| < ∞ and one often says that the process has short memory, although it

might be preferable to call it “medium” memory, since exponentially decaying correlations

might better describe “short” memory.

More details on fBm can be found in Nualart’s textbook [20].

2.2 Fractional Ornstein-Uhlenbeck Process

The fractional Ornstein-Uhlenbeck process is the fractional analogue of the well-known Ornstein-

Uhlenbeck process; it is a continuous-time first-order autoregressive process X = {Xt; t ≥ 0}which is the solution of an one-dimensional homogeneous linear stochastic differential equation

driven by a fBm {BHt ; t ∈ R+} with Hurst parameter H ∈ [1/2, 1). Specifically, it is the unique

Gaussian process satisfying the following linear stochastic integral equation

Yt = α

∫ t

0Ys ds + β BH

t , (4)

where α and β are constant drift and variance parameters, respectively. The solution to

Equation (4) is stationary, almost surely continuous, and H-self-similar. The decay of the

autocovariance function of {Yt; t ∈ R+} is similar to that of the increments of the fBm, and

thus it exhibits long-range dependence. See Cheridito et al. [6] for more details.

3 Option Pricing under the long memory stochastic volatility

model

Although the model we consider is in continuous-time, in practice what we have available are

discrete-time observations of historical stock prices St1 , St2 , . . . , StK , where tK is the time

today. In particular, we cannot hope to obtain observations of S of arbitrarily high frequency,

and to make matters worse, the volatility process Y is unobserved. In order to deal with the

option pricing problem under these circumstances, we start by estimating the filtered stochastic

volatility distribution given historical stock price observations; we do this by using the so-called

stochastic volatility particle filter. Then, we use a multinomial recombining tree to compute

the price of the option.

6

3.1 Empirical Distribution of the Volatility Process

In this section we deal with historical stock prices, thus we work under the objective probability

measure P, which is the measure that corresponds to all past observations. Let

pi(dy) := P[Yti ∈ dy|Xt1 , . . . , Xti ].

This is a probability measure for each i = 1, . . . , K, where K is the number of historical

observations available. Since Xt1 , . . . , Xti are random variables, it is customary to consider

that pi(dy) is random since it depends on these, and a common appellation for it consistent

with this point of view is the stochastic filter of Y given discrete values of X. However, when

Xt1 , . . . , Xti are observed, it is better to consider that pi is non-random at time i, as it depends

entirely on these i known values at that time.

We now explain how to construct n time-varying particles {Y jti

: i = 1, . . . , K, j =

1, . . . , n} together with their corresponding probabilities {pji : i = 1, . . . , K, j = 1, . . . , n}, such

that given the observations Xt1 , . . . , Xti , the empirical distribution of the particles converges

to the probability measure pi(dy) as n increases to infinity.

To this end, we apply a genetic-type algorithm with two steps: a mutation step and a

selection step. We refer to the article [10] by Del Moral et al. for a mathematical analysis of the

method, and to [2] and [13] for its implementation in option pricing and portfolio optimization

problems with no memory. The following is a description of the algorithm adapted to our

context:

Step 0 :

Assume that we have K +1 past price observations at times t0 < t1 < . . . < tK , where tK

is the time today (the time of the most recent observation). For convenience we assume

that these times are equidistant: ti − ti−1 = ∆t = h. Pick a bounded integrable function

φ from Rq into [0,∞) such that∫

φ(x)dx = 1, and

∫

|x|φ(x)dx < +∞.

We choose to work with φ(x) = e−2|x|. For n > 0, define also the contraction φn(x) =3√

nφ(x 3√

n). The power 1/3 is taken to be consistent with the convergence theorem of

del Moral et al., but in practice we have not noticed any sensitivity to other moderate

choices of powers. Start by dividing the first interval [t0, t1] into M pieces as follows:

t0 = s0 < s1 < . . . < sM = t1.

In the sequel, M will be the number of Euler steps in each time interval [ti−1, ti].

7

Iteration # 1 :

We initialize our iteration scheme for the particles representing X and Y :

• Xt0 = x0, where x0 is the observed value of X from the historical prices at time t0.

• Yt0 = y0, where y0 can be an arbitrary point or a value sampled from the stationary

distribution of Y or an estimate of Y based on the historical data, such as the

historical volatility or the implied volatility.

Mutation Step:

1. We simulate M values of fBm, denoted by

(

BH,jt0

(s0), BH,jt0

(s1), . . . , BH,jt0

(sM ))

.

See comment in Remark (i) below on how to construct these values.

2. We simulate recursively the SDEs in (2) using an Euler scheme for k = 0, . . . ,

M − 1. For convenience denote Y jt0

(sk) = Y jt0, k and Xj

t0(sk) = Xj

t0, k; we set

Y jt0, k+1 = Y j

t0, k + α Y jt0, k(sk+1 − sk) + β

(

BH,jt0

(s0) − BH,jt0

(s0))

Xjt0,k+1 = Xj

t0, k +

µ −σ2

(

Y jt0,k

)

2

(sk+1 − sk) + σ(

Y jt0,k

)

Zk

√

sk+1 − sk

where Zk are independent standard normal random variables.

3. We keep only the final values from the previous recursive step

Y jt1

:=Y jt0,M = Y j

t0(sM )

Xjt1

:=Xjt0,M = Xj

t0(sM )

We repeat Substeps 1 to 3 above, n times independently for each j, so in the end

of the mutation step we have constructed the pairs {Xjt1

, Y jt1}j=1,...,n.

Selection Step:

Introduce the following discrete probability measure constructed from the pairs at

the end of the mutation step

Φn1 (dy) =

1C

∑nj=1 φn

(

Xjt1− x1

)

δ{Y jt1}, if C > 0

δ{0}, otherwise

8

Those simulated particles which are closer to the observed return value of x1 =

log S(t1) will have a higher weight, because φn acts as an approximation of the

dirac delta function. The constant C is chosen such that Φn1 (dy) is a probability

measure: C :=∑n

j=1 φn

(

Xjt1− x1

)

.

Iterations # 2 to K :

For each iteration i = 2, 3, . . . , K we apply the Mutation step above with initial values

for the Euler scheme, the observation xti−1, which is known at time ti−1 and the volatility

value Y jti−1

sampled from the distribution Φni−1; each sample is done independently for

each j, but the sampling distribution depends on the entire particle system at the previous

step, which is why this algorithm is considered an interacting particle system. Thus, in

the end of the mutation step we obtain n pairs {Xjti, Y j

ti}j=1,...,n. Then, we apply the

Selection step and we obtain

Φni (dy) =

1C

∑nj=1 φn

(

Xjti− xi

)

δ{Y jti}, if C > 0

δ{0}, otherwise

where C :=∑n

j=1 φn

(

Xjti− xi

)

.

Output :

At each step i, the discrete distribution Φni is the estimate of pi(dy), i.e. of the distribution

of Yt at ti, given the observed stock prices x1, x2, . . . , xi.

Some Remarks:

(i) In order to produce the fBm observations needed in the Euler Mutation step we can

choose any simulation method from the existing ones. We refer to [11] for a detailed

exposition of all the techniques. Here, we choose to work with the circulant method,

since it is faster than others (of order n log n).

(ii) In order to simulate the SDE that describes X, it is well-known that an equidistant Euler

discretization converges to the continuous-time SDE. For Y this is not trivial. However,

based on the work by A. Neuenkirch (2006, [22]), we can apply an equidistant Euler

scheme, in order simulate the SDE recursively and get convergence to the continuous-

time fractional SDE. However, the convergence of the Euler scheme of the fractional SDE

is slower than that of geometric Brownian motion and depends on the value of H; the

higher the value of H the slower the convergence to the continuous-time model. Thus,

unless H is very close to 1/2, it is better to work with a relatively large number of Euler

steps to guarantee convergence, for example M = 600.

9

3.2 Multinomial Recombining Tree

From the previous section, using past information of the stock, we have constructed the discrete

distribution ΦnK of the empirical volatility and this will serve as our estimate of the present

volatility. We now abandon the notation tK for today’s time stamp, in favor of using time 0.

Today’s stock price is thus S0, which is observed. We wish to price an option with maturity

T ; thus our time interval is [0, T ].

Our tree consists of N periods, which means that we divide [0, T ] into N equidistant

subintervals: ∆t = TN

. The tree is branching at times i∆t. The nodes of the tree correspond

to the return values of the stock, i.e. Xt = log St.

The procedure below is described for the ith timestep. Let the return be denoted by x.

Then successors together with their corresponding probability weights are computed following

Florescu and Viens, [13], as follows:

Step 1: Sample a volatility value from the particle filter distribution, ΦnK ; denoted by Yi.

Step 2: Let j =[

x

σ(Yi)√

∆t

]

+1, where [·] denotes the integer part, and take the four successors

of x to be

x1 = (j + 1)σ(Yi)√

∆t +(

r − σ2(Yi)s

)

∆t

x2 = jσ(Yi)√

∆t +(

r − σ2(Yi)s

)

∆t

x3 = (j − 1)σ(Yi)√

∆t +(

r − σ2(Yi)s

)

∆t

x4 = (j − 2)σ(Yi)√

∆t +(

r − σ2(Yi)s

)

∆t

In order to compute the corresponding weights of the successors, let

δ =

{

x − jσ(Yi)√

∆t, if |(x − jσ(Yi)√

∆t)| < |x − (j − 1)σ(Yi)√

∆t|x − (j − 1)σ(Yi)

√∆t, otherwise

(5)

and let q = δ/(

σ(Yi)√

∆t)

be the standardized distance. Now depending on the value

of q we have two different sets of probability weights, as follows:

- If q ∈ [−12 , 0] the probability weights are given by

p1 = 12(1 + q + q2) − p, p2 = 3p − q2

p3 = 12(1 − q + q2) − 3p, p4 = p.

(6)

- If q ∈ [0, 12 ] then

p1 = p, p2 = 12(1 − q + q2) − 3p

p3 = 3p − q2, p4 = 12(1 + q + q2) − p.

(7)

10

Remark 1: If we construct a one-step quadrinomial tree with the successors given above

where p is the probability of the successor furthest away from x, then p ∈[

112 , 1

6

]

and in

both cases above, the relations (6) and (7) define an equivalent martingale measure. Since

p is not uniquely determined, this implies that our martingale measure is not unique. We

refer to [13], and to point (i) in Remark 3 below, on how to choose a value of p which is

consistent with the market.

Step 3: To construct the multi-period model we start by sampling n values from the discrete

volatility distribution Φni . Then, with an initial value x0 we start by computing the four

successors of x0 and their corresponding weights, as in Steps 1 and 2 for the first sampled

value Y1. After this, for each of one of the four successors we compute their respective

successors for the second sampled volatility value Y2 and so on.

Step 4: Once we have constructed the tree we can use a standard pricing technique that is

consistent with the no-arbitrage condition: we compute the value of the payoff function

at the terminal nodes and we continue using a backward induction algorithm in order

to compute the option price at time 0, as the discounted expecation of the final nodes

using the martingale measure computed in Step 2. Since the tree is recombining by

construction, the level of computation is polynomial, and we have found in practice that

is of order no greater than Nα for some α < 3.

Step 5: We can iterate this entire procedure by using n′ repeated samples {Y 1, . . . , Y n′}from Φn

K , constructing a different tree with each sample, and then using a Monte-Carlo

approach to average all prices obtained from each generated tree using each sample. This

last step reduces variability due to the single sampling used to construct the tree, and is

consistent with the fact that the arbitrage-free pricing can be represented using expected

values. It results in an algorithm which approximates the standard arbitrage-free price

of the claim under the martingale measure chosen in step 2.

Remark 2: Since we price the option using a tree, it works for both American and

European options.

3.3 Simulated Data Example

We simulate 255 datapoints (corresponding to one year of data) with initial values x0 = 6.802

and y0 = 0.35. The chosen parameters for the model are: H = 0.6, α = 0.027, β = 0.076,

µ = −0.0014 and σ(y) = y. The simulated model is shown in Figure 1 (i).

11

Simulated LOG−Stock Prices

Days

Sto

ck P

rices

0 50 100 150 200 250

6.8

7.0

7.2

Simulated Underlying Volatility (H=0.6)

Days

Vol

atili

ty

0 50 100 150 200 250

0.35

00.

356

0.36

2Estimated Discrete Volatility Distribution

Volatility

Freq

uenc

y

0.358 0.359 0.360 0.361 0.362 0.363

010

2030

4050

(i) (ii)

Figure 1: (i) Simulated model, (ii) Empirical Discrete Volatility Distribution of the simulated

model.

Using the particle filter algorithm described in Section 3.1 we construct the discrete

empirical distribution of Y , using M = 600 Euler steps and n = 1000 particles. The particle

filter is shown in Figure 1 (ii).

We use the multinomial tree algorithm described above in order to compute the price

of a European call option with initial stock price S0 = 1248.413, risk-free interest rate r = 0.02

and maturity T = 100 days. In the tree we use N = 50 time steps and we choose p = 0.16.

The option prices computed for a range of strike prices are shown on Table 1 together with

the corresponding prices computed using a Binomial and a Black-Scholes model.

Remark 3:

(i) We compute option prices for various values of the free parameter p and we plot these

prices with respect to p. As shown in Figure 2 (i) the computed option prices are robust

with respect to the choice of p.

(ii) In Figure 2 (ii) we can visualize the level of recombination of the tree which is quite high.

Actually, there is a linear growth of nodes for each additional step.

4 Inference for the Long-Memory Stochastic Volatility Model

Before applying our technique to real-time data, it is crucial to discuss how we can estimate

the parameters in the model. These are: the mean rate of return µ, the parameters α and β

of the fractional SDE and the memory-parameter H.

12

Strike Price BS Price Binomial Price SV Price

700 554.1863303 554.1849152 554.1057391

750 505.0754056 505.0731263 505.0367213

800 456.5147933 456.5091821 456.5332716

850 408.8715749 408.8625159 408.9530273

900 362.6076577 362.604282 362.7570372

950 318.2430249 318.2361917 318.439265

1000 276.3039653 276.2815377 276.5298347

1050 237.2679239 237.2904777 237.4716566

1100 201.5163164 201.5084491 201.6655051

1150 169.3031044 169.3418959 169.3787487

1200 140.7421204 140.7419948 140.7180868

1250 115.8118004 115.7890584 115.6567938

1300 94.37315462 94.38510372 94.09626885

1350 76.19566378 76.21586201 75.81583452

1400 60.98603152 60.99854268 60.51299286

1450 48.41578386 48.40507526 47.86828359

1500 38.14508725 38.10619578 37.57190267

1550 29.84146784 29.84695049 29.24575048

1600 23.19315611 23.20979316 22.60585766

1650 17.91748433 17.90359371 17.34007038

1700 13.76515254 13.75997314 13.23094349

1750 10.52132021 10.52142996 10.03311333

1800 8.004455588 7.988044783 7.565121012

1850 6.063753329 6.060815433 5.674675782

Table 1: European call option prices computed for the simulated model for a range of strike

prices for the simulated model stochastic volatility, the Black-Scholes and the Binomial model.

13

0.10 0.12 0.14 0.16

8010

012

014

016

0Option Price vs. p

Probability

Opt

ion

Pric

e

BS PriceSV Price

0 100 200 300 400 500 600

050

000

1500

0025

0000

3500

00

Rate of Growth in the Tree

Number of Steps

Tota

l No

of N

odes

in th

e Tr

ee

Binomial growthStatic TreeLinear growthn^(1.5)n^(1.75)

(i) (ii)

Figure 2: For the simulated model: (i) Robustness with respect to p , (ii) Growth with respect

to N .

It would be optimal to estimate these parameters jointly. Unfortunately, such a problem

is highly non-trivial and to our knowledge has not been solved even in the case in which we

actually observe a fractional Ornstein-Uhlenbeck process. In the majority of the state-space

models with long-memory in the literature, a rigorous way to jointly estimate this vector of

parameters is unknown. The difficulty is mainly due to the fact we do not know H. If H were

known, then using high-frequency data we could adjust standard techniques in the literature

for non-fractional mean-reverting stochastic volatility models in order to estimate α and β.

For example, one can use the variogram approach as discussed in Section 4.2 in Fouque et al.,

[14]. Therefore we begin with a discussion of how to find H.

4.1 Inference for the long-memory parameter H

If we want to estimate H, the first issue that arises is that the Hurst index appears in the

volatility process which is unobserved. However, as discussed in several articles in the literature

[9], [19], evidence of long-memory appears in the squared log-returns. More specifically, if Xt =

log St+1 − log St, then Cov(Xt+1, Xt) = 0 (which complies with our hypothesis of uncorrelated

returns), but

Cov(X2t+1, X

2t ) ≈ t2H−2

Therefore, we could apply a non-parametric technique in order to estimate H. The best

statistic to estimate H is the Rescaled-Range statistic (R/S statistic); it is non-parametric.

We implement it in the simulated example discussed in the previous section. The true value of

14

Method H (true H = 0.6) st. error(H)

R/S Statistic 0.88 0.02

Periodogram 1.59 0.23

Boxed (or modified) Periodogram 1.43 0.05

Peng’s (or Variance of residuals) 1.29 0.04

Higuchi’s (or fractal dimension) 0.94 0.04

Table 2: Various non-parametric methods for estimating H from the squared log-returns and

their corresponding standard errors.

H in our simulated model is 0.6; using the R/S statistic in the squared returns, we obtain is

H = 0.88 with corresponding standard error se(H) = 0.02. We also applied the same procedure

for various simulated models and the results were similarly unsatisfactory. The situation is

the same when we estimate H using other non-parametric techniques such as the aggregated

variance or the periodogram method. The estimated values of H using various non-parametric

techniques are summarized in Table 2.

This leads to the following natural question: how important is it to estimate H prop-

erly? That is, if a rough estimate of H does not affect significantly the discrete distribution

of Y and consequently the option price, then its estimation is of little consequence. To check

the effect of H in the particle filter as well as the option price, we need perform a sensitivity

analysis with respect to H. Before doing so, we study the effect of the number of particles

chosen on the discrete volatility distribution estimated from the particle filtering algorithm.

4.1.1 Empirical Volatility Distribution & Number of Particles

In [13], H = 0.5 was used (that is the volatility is driven by a standard Brownian motion

and thus the model is Markovian), the number of particles needed for the particle filtering

algorithm was not large. Obviously, by using a large number of particles, for example n = 1000,

we generate a finer mesh for the discrete volatility distribution, finer than the 20 or so particles

needed to get satisfactory results as in [13], but the empirical range of the volatility filter or

the resulting option price remains of the same magnitude whether one uses 20 or 1000 particles

in this short memory case. However, this is not the case when H is greater than 0.5. One can

easily see (for example in Figure 3) that if we keep the number of particles constant, then the

very nature of the generated particle filter depends on the value of H, in the sense that the

higher H is the smaller the spread of the empirical distribution is.

The plots in Figure 3 imply that we need to use a significantly larger number of particles

15

H=0.50

volatility

Fre

qu

en

cy

0.220 0.240

01

5

H=0.51

volatility

Fre

qu

en

cy

0.220 0.240

08

H=0.52

volatility

Fre

qu

en

cy

0.220 0.240

01

5

H=0.53

volatility

Fre

qu

en

cy

0.220 0.240

01

5

H=0.54

volatility

Fre

qu

en

cy

0.220 0.240

06

H=0.55

volatility

Fre

qu

en

cy

0.220 0.240

01

5

H=0.56

volatility

Fre

qu

en

cy

0.220 0.240

01

5

H=0.57

volatility

Fre

qu

en

cy

0.220 0.240

08

H=0.58

volatility

Fre

qu

en

cy

0.220 0.240

06

H=0.59

volatility

Fre

qu

en

cy

0.220 0.240

08

H=0.60

volatility

Fre

qu

en

cy

0.220 0.240

08

H=0.61

volatility

Fre

qu

en

cy

0.220 0.240

01

5

H=0.62

volatility

Fre

qu

en

cy

0.220 0.240

01

5

H=0.63

volatility

Fre

qu

en

cy

0.220 0.240

06

H=0.64

volatility

Fre

qu

en

cy

0.220 0.240

08

H=0.65

volatility

Fre

qu

en

cy

0.220 0.240

01

5

H=0.66

volatility

Fre

qu

en

cy

0.220 0.240

01

5

H=0.67

volatility

Fre

qu

en

cy

0.220 0.240

06

H=0.68

volatility

Fre

qu

en

cy

0.220 0.240

01

5

H=0.69

volatility

Fre

qu

en

cy

0.220 0.240

06

H=0.70

volatility

Fre

qu

en

cy

0.220 0.240

08

H=0.71

volatility

Fre

qu

en

cy

0.220 0.240

06

H=0.72

volatility

Fre

qu

en

cy

0.220 0.240

08

H=0.73

volatility

Fre

qu

en

cy

0.220 0.240

01

5

H=0.74

volatility

Fre

qu

en

cy

0.220 0.240

01

5

Figure 3: Histogram of the estimated discrete volatility distribution for the IBM stock

(07/22/2009 - 08/24/2009) for different values of H keeping constant the number of parti-

cles and equal to n = 50.

16

Histogram of the Volatility Filter

Volatility (in years)

Fre

quen

cy

0.2190 0.2195 0.2200

02

46

810

1214

Histogram of the Volatility Filter

Volatility (in years)

Fre

quen

cy

0.22030 0.22035 0.22040 0.22045 0.22050 0.22055 0.22060

05

1015

H = 0.6 H = 0.72

Figure 4: Histogram of the estimated discrete volatility distribution of the simulated model

for different values of H.

in order to keep the spread of the empirical distribution roughly constant. By performing

several simulations using different models, we observe that if we use 1000 particles or more,

then the spread of the particle filter for all the values of H is robust (it remains roughly the

same for all values of H and for all filters with over 1000 particles ). The only disadvantage

now is that the time required to generate such a filter increases significantly. Note however that

for pricing problems, the filter can be computed off-line and updated as the last observations

arrive thereby significantly reducing the computational time at the time of pricing.

4.1.2 Sensitivity Analysis with respect to H

We use the same simulated model as in Section 3.3, in which the true value of H was 0.6,

and we plot the price of a European call option computed from values of H between 0.5 and

0.95 with step 0.05. For the remaining parameters, we use the true values, but we increase the

number of particles as H is closer to 0.95, so that we keep the range of the generated histogram

roughly constant. Then, for each particle filter corresponding to different values of H we use

the multinomial tree algorithm in order to compute the value of the option and then we plot

it against H.

In general, in the different particle filters generated for the various values of H for the

simulated model, we observe that as H gets closer to 1, the histogram is shifted to the right,

meaning that the volatility tends to obtain higher values. Furthermore, in certain cases there

is no overlap between the range of the volatility values.

17

0.50 0.55 0.60 0.65 0.70 0.75

24.2

024

.25

24.3

024

.35

24.4

024

.45

24.5

0

Option Prices vs. Hurst Index

Hurst index

Opt

ion

pric

e

Bid−Ask SpreadBS

Figure 5: European call option prices with respect to different values of H for the simulated

model and comparison with BS.

To continue our sensitivity analysis we plot the option prices obtained for against

different values of H; this is shown in Figure 5.

In the graph we also compare the option price obtained using our long-memory stochas-

tic volatility model with a standard Black-Scholes option price, in which the volatility used

is the historical volatility. Moreover, we price an option which is at the money (i.e. Strike

price K = 1250). We observe that the option price that is closer to the BS price is the one for

H = 0.6 while the rest diverge significantly from it. In general, whether or not we are compar-

ing our output with BS prices, we see that the effect of H on the option price is significant.

We must conclude that its proper estimation is crucial.

4.2 Implied H approach

Section 4.1.2 shows it is very important to estimate H properly, since the value of the option

depends on the true value of H. In addition, from what we discussed in Section 4.1, there is no

known parametric or non-parametric approach to estimate H in a satisfactory way. However,

a closer inspection of Figure 5 shows that our data can give us information on the underlying

value of H and this can be done using an implied approach similar to that of the implied

volatility technique.

The idea is to use our data and generate various volatility filters for a range of values

18

of H and then use the realized option prices in order to obtain a value of H that most closely

matches our data. The range of values of H should be between 0.5 and 0.95 and the step

should be 0.05 or smaller. If we use a step size of 0.01, then the differences in the computed

option prices will be quite small and several values of H will be close to the true option price

(in the sense that they are inside the bid-ask spread). Therefore we need a standardized way

to choose H. Towards this direction, we suggest using the Mean Square Error (MSE) between

the computed option price and the center of the bid-ask spread. Then, the chosen value of

implied H would be the one that corresponds to the smallest MSE. We formalize our technique

in the steps:

1. For values of H varying from 0.5 to 0.95 (with increments of 0.01):

(a) Estimate the filtered stochastic volatility distribution.

(b) Compute the corresponding option prices for various strike prices.

(c) Compute the mean squared error of the option price from the center of the market’s

bid-ask spread for that same option.

2. Choose the value of H that corresponds to the smallest MSE.

Once we have chosen the implied value of H, i.e. once our model has been calibrated

to the market data, we can use it to generate a volatility particle filter which we then use in

our multinomial recombining tree algorithm. One drawback is the computational time needed

to generate all these different particle filters (especially for higher values of H). However, this

is not a major issue since this calibration procedure does not have to be repeated every time

we compute an option price, but only once, for at the beginning of each trading day.

5 Application to Market Data

We apply our methodology to pricing European call options on the S&P 500, investigating at

the same time whether there is long-memory (i.e. H > 0.5) in the volatility or not.

Remark 4: The historical stock prices we use to estimate the empirical volatility ditri-

bution are obtained from Yahoo Finance. The corresponding option prices are obtained from

the “Delta Neutral” database, and the interest rates from the US Department of the Treasury.

For the first example we use 1 month’s data, between December 29, 2008 and January

29, 2009 to estimate the volatility particle filter. The day “today” is January 30, 2009 and we

wish to price a European call option on the S&P 500 with expiration on March 20, 2009. The

19

underlying stock price is $825.88 and the strike price we consider is K = $820. The implied

volatility (for an at-the-money option) is σimplied = 0.405, while the historical volatility is

computed as σhistorical = 0.387. The risk-free interest rate is r = 0.24% (we consider as

interest rate the yield of a 3-month maturity T-bond).

The first task is to estimate the parameters of the model. Using the approach described

in the previous section we find:

α = 0.0585, µ = −0.0054.

The next task is to compute the implied value of H in order to price the option we

are interested in. As discussed above, we generate various volatility filters. We choose to work

with M = 600 Euler steps and n = 1000 particles. In order to compute the theoretical option

prices that will be used for calibration we consider a tree of N = 100 steps and we use a

Monte-Carlo approach by generating m = 1000 trees and averaging over the different option

prices.

In Table 3 we can see a sample from the computed option prices using the long-memory

stochastic volatility that we just described and the corresponding bid-ask spreads. It turns out

that the value of H that gives us the smallest MSE is H = 0.53.

A first observation from our results in Table 3, where our prices are compared to the

bid-ask spread, is that our model tends to underestimate options that are deep in the money

and overestimate those that are out of the money, while for options at the money the computed

option price is very close to the center of the bid-ask spread.

Remark 5: One crucial issue to discuss is the time horizon over which we consider H

to be constant. This is important because it determines the range of the historical data that we

are going to use in order to generate the volatility filter as well as the maturity of the options

we wish to price. We use market data from the S&P 500 from different time periods and of

various time ranges. More specifically we consider data from January 2008 until December

2008 and from January 2009 until August 2009. The time horizons we study, and the maturity

of the options we price, are 1 month, 2 months, 3 months, 6 months and 1 year. It turns out

that it is quite safe to consider H to be constant for a 2-month period. Therefore, using 1

month historical data, we are able to price option that expire in one month from today.

From the first example we observe that there is evidence of long-memory in the S&P 500

data. However, it would be interesting to investigate whether this observation is something

systematic or not. We choose to work with data from the 2008 Wall Street crisis for two

different periods: (a) between March 7th and May 9th 2008 and (b) between September 12th

and December 19th 2008. the first period corresponds to the collapse of the Bear Stearns

20

Strike Price Bid Ask H=0.5 H=0.51 H=0.53 H=0.55 H=0.57 H=0.59

700 138.2 140.8 131.73 131.46 131.33 131.43 131.63 131.20

710 129.8 132.3 123.45 123.18 123.06 123.15 123.36 122.93

720 121.4 123.9 115.41 115.14 115.03 115.12 115.31 114.90

730 113.3 115.8 107.63 107.35 107.25 107.33 107.53 107.12

740 105.3 107.8 100.11 99.84 99.75 99.82 100.02 99.62

750 97.5 100 92.88 92.61 92.53 92.60 92.79 92.41

760 89.9 92.4 85.95 85.68 85.60 85.67 85.86 85.49

770 82.6 85 79.32 79.05 78.99 79.05 79.23 78.88

780 75.5 77.8 73.02 72.74 72.69 72.74 72.92 72.58

790 68.6 71 67.02 66.75 66.70 66.76 66.93 66.60

800 61.9 64.4 61.35 61.09 61.05 61.10 61.26 60.95

810 55.6 58.2 56.01 55.75 55.71 55.76 55.92 55.62

820 49.5 52.3 50.98 50.74 50.71 50.75 50.90 50.62

830 43.8 46.7 46.28 46.04 46.02 46.06 46.20 45.94

840 38.6 41.4 41.89 41.66 41.65 41.68 41.82 41.57

850 33.5 36.4 37.82 37.60 37.59 37.61 37.74 37.51

860 28.9 31.8 34.04 33.83 33.82 33.85 33.97 33.76

870 24.7 27.5 30.56 30.36 30.36 30.38 30.49 30.29

880 20.7 23.5 27.36 27.16 27.16 27.18 27.29 27.11

890 17.8 20.0 24.42 24.24 24.24 24.26 24.36 24.19

900 14.7 16.8 21.74 21.57 21.57 21.59 21.68 21.52

Table 3: Implied H: Option prices for different values of H.

21

Companies and the second one to the crash of September 2008 with the bankruptcy of Lehman

Brothers, Inc. and the “bailout” of American International Group, Inc.

Bear Stearns Companies, Inc. In this example we want to study whether there is evidence

of long-memory in the market after the crash of the Bear Stearns.

We price European call options on the S&P 500 index with maturity approximately one

month and using historical data of one month’s range. Using a rolling window of a week,

we price a European call option on Friday, March 7th 2008 with expiration in a month

(based on historical data between 02/01/08 until 03/06/08) and we continue until Friday,

May 9th 2008 with an option that expires in a month (based on historical data between

03/01/08 until 04/08/08).

Figure 6 has the values of H that we computed using our implied technique for each

week. We observe that that there is no structural long-memory in the stock and options

market before the crash, since the value of Himplied is 0.5. However, the week after the

crash, it appears that a model with long-memory in the volatility is more suitable for the

S&P 500, since Himplied is higher than 0.5. The memory in the market is not persistent,

since one month after the investment bank’s collapse the implied value of H returns to

0.5 and remains there.

Lehman Brothers Holdings, Inc. & American International Group, Inc. (AIG) One

might expect that the full-blown financial crisis of September 2008 had a stronger effected

on the memory of the market than the failiure of a single investment bank. Indeed,

Lehman Brothers declares bankruptcy on Monday, September 15, in one of the largest

bankruptcy filings in the U.S. history. One day later, on September 16 the American

International Group, Inc. (AIG) receives a $85 billion credit facility from the U.S. Fed-

eral Reserve Bank to meet increased collateral obligations consequent to the credit rating

downgrade in the beginning of same month. This crisis created instability, resulting in

the Dow Jones reaching a 6-year low on November 20.

As in the previous example, we price European call options on the S&P 500 index with

maturity approximately one month and using one-month historical data. We compute

prices beginning on Friday, September 9, 2008, based on historical data from 09/08/08

going back to 08/01/08. We continue pricing until Friday, December 19, 2008 for an op-

tion that expires in a month (based on historical data between 11/01/08 until 12/18/08).

In Figure 7 we plot the values of H for each week.

22

0.50

0.51

0.52

0.53

0.54

Bear Stearns Companies, Inc.

Date

Hur

st In

dex

03/0

7/08

03/1

4/08

03/2

1/08

03/2

8/08

04/0

4/08

04/1

1/08

04/1

8/08

04/2

5/08

05/0

2/08

05/0

9/08

Figure 6: Bear Stearns Companies, Inc. was a global investment bank, securities trading and

brokerage firm. On March 16 , 2008, Bear Stearns signed a merger agreement with JP Morgan

Chase in a stock swap worth $2 a share or less than 10% of its market value just two days

earlier.

23

Although the market is quite unstable during this period of time, there is no evidence

of long-memory before the crash, since the value of Himplied is 0.5. However, after the

crash, the value of H is higher than 0.5 and remains high for an extended period (more

than three months). This is evidence that the amplitude of the information hitting the

market is related to its persistence over time; it also suggests that long memory may be

an aspect of market instability.

6 Conclusion

In this article we studied a long-memory stochastic volatility model: we introduced a method-

ology for option pricing under such a model as well as a parameter estimation procedure.

The determination of the market’s current volatility structure is based on the construction

of largely non-parametric particle filter for the empirical distribution of the volatility using

historical prices of the underlying. This empirical distribution is the basis of a multinomial

recombining tree, used for arbitrage-free option pricing. The option pricing algorithm has the

advantage of employing a tree structure, which implies that it can be used for pricing any

path-dependent option. In addition, since the tree is highly recombining, by construction, the

computational time needed for pricing is minimized.

The construction of the particle filter is time-consuming; and this is compounded by

the fact that we we need to compute the implied value of H, since several particle filters (for

different values of H) need to be computed. However, this is not practical issue, since the filters

can be computed offline based on historical data, for instance at the start of each trading day;

H may need to be calibrated even less frequently, since one observes that it remains constant

for a month at a time.

Since the entire option pricing scheme is found to be sensitive to the value of H, and

standard long memory estimation techniques are found to be ineffective for stock price data,

even in simulations, we suggest an implied technique to extract H by minimizing the Mean

Square Error of the distance between the computed option price (using our algorithm) and

the center of the bid-ask spread given by the options market. The fact that in the market

data examples the MSE is minimized for values of H higher than 0.5 also implies that the

prices computed under the long-memory volatility model were closer to the center of the bid-

ask spread than under the “classical” memoryless volatility model. This is of some practical

importance, since we chose our examples from periods of high market instablility.

The question of the construction of a hedging strategy to accompany our pricing scheme

is an ongoing project and will be discussed in a future article.

24

0.50

0.51

0.52

0.53

0.54

0.55

0.56

Lehman Brothers Holdings Inc. & American International Group (AIG)

Date

Hur

st In

dex

09/1

2/08

09/1

9/08

09/2

6/08

10/0

3/08

10/1

0/08

10/1

7/08

10/2

4/08

10/3

1/08

11/0

7/08

11/1

4/08

11/2

1/08

11/2

8/08

12/0

5/08

12/1

2/08

12/1

9/08

Figure 7: Lehman Brothers Holdings Inc. was a global financial services firm which, un-

til declaring bankruptcy in 2008, participated in business in investment banking, equity and

fixed-income sales, research and trading, investment management, private equity, and private

banking. It was a primary dealer in the U.S. Treasury securities market. American Interna-

tional Group, Inc. (AIG) is an American insurance corporation. AIG was once the 18th-largest

public company in the world. It was listed on the Dow Jones Industrial Average from April 8,

2004 to September 22, 2008.

25

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